Theorem crnggrpd 20324

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20323	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20319	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2143  Grpcgrp 18995  CRingccrg 20311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-ring 20312  df-cring 20313

This theorem is referenced by:  fermltlchr  21679  selvvvval  22293  selvadd  22294  psdmul  22329  psd1  22330  psdpw  22333  ply1chr  22466  ply1fermltlchr  22472  elrgspnsubrunlem1  33567  elrgspnsubrunlem2  33568  erlbr2d  33584  rlocaddval  33589  rloccring  33591  rloc0g  33592  rlocf1  33594  fracerl  33627  gsumind  33665  dflringlem2  33785  ressply1evls1  33855  evl1deg1  33866  evl1deg2  33867  evl1deg3  33868  ply1dg1rt  33870  vr1nz  33883  mplasclco  33906  psrmonprod  33942  mplmonprod  33944  esplyfvn  33967  irngss  34077  extdgfialglem1  34082  irredminply  34106  algextdeglem4  34110  algextdeglem5  34111  aks6d1c1p3  42877  aks6d1c2lem4  42894  aks6d1c6lem2  42938  aks5lem2  42954  evlsbagval  43318  evlselv  43321  prjcrv0  43365